The article provides an overview of recent progress in understanding the dynamics of closed quantum many-body systems out of equilibrium, focusing on dynamical equilibration and thermalization. It discusses various scenarios such as quenches, ramps, and periodic driving, and addresses topics like the eigenstate thermalisation hypothesis, typicality, transport, many-body localization, and universality near phase transitions. The review highlights the importance of local quantum many-body systems with finite-range interactions, exemplified by spin lattice models like the Ising model and Bose-Hubbard models. It explains how these systems relax to a maximum entropy state after a quench, and how this relaxation can be rigorously proven for certain models. The article also explores the concept of thermalisation, including the eigenstate thermalisation hypothesis and the role of integrability in preventing full thermalisation. Additionally, it discusses the dynamics of quantum phase transitions, the impact of disorder on transport properties, and the potential of quantum simulations using ultra-cold atoms and trapped ions. The review concludes by outlining future perspectives, emphasizing the rapid development of the field and the need for further theoretical and experimental advancements.The article provides an overview of recent progress in understanding the dynamics of closed quantum many-body systems out of equilibrium, focusing on dynamical equilibration and thermalization. It discusses various scenarios such as quenches, ramps, and periodic driving, and addresses topics like the eigenstate thermalisation hypothesis, typicality, transport, many-body localization, and universality near phase transitions. The review highlights the importance of local quantum many-body systems with finite-range interactions, exemplified by spin lattice models like the Ising model and Bose-Hubbard models. It explains how these systems relax to a maximum entropy state after a quench, and how this relaxation can be rigorously proven for certain models. The article also explores the concept of thermalisation, including the eigenstate thermalisation hypothesis and the role of integrability in preventing full thermalisation. Additionally, it discusses the dynamics of quantum phase transitions, the impact of disorder on transport properties, and the potential of quantum simulations using ultra-cold atoms and trapped ions. The review concludes by outlining future perspectives, emphasizing the rapid development of the field and the need for further theoretical and experimental advancements.